sigma: a new open economy model for policy analysis - …henderson, ben hunt, steve kamin, doug...

SIGMA: A New Open Economy Model for

Policy Analysis∗

Christopher J. Erceg, Luca Guerrieri, and Christopher GustFederal Reserve Board

In this paper, we describe a new multicountry open eco-nomy SDGE model named “SIGMA” that we have developedas a quantitative tool for policy analysis. We compareSIGMA’s implications to those of an estimated large-scaleeconometric policy model (the FRB/Global model) for anarray of shocks that are often examined in policy simulations.We show that SIGMA’s implications for the near-term res-ponses of key variables are generally similar to those ofFRB/Global. Nevertheless, some quantitative disparitiesbetween the two models remain due to certain restrictiveaspects of SIGMA’s optimization-based framework. We con-clude by using long-term simulations to illustrate some areasof comparative advantage of our SDGE modeling framework.

JEL Codes: E32, F41.

In the wake of the Lucas critique and the early real business cycle(RBC) literature, a wide gap emerged between the models exam-ined in the academic literature and those utilized in policy analysisby most central banks. While central bank policy models retained

∗We appreciate comments and suggestions from Tamim Bayoumi, RalphBryant, Pablo Burriel (discussant), Matthew Canzonieri, Lawrence Christiano,Behzad Diba, Martin Eichenbaum, Joe Gagnon, Jordi Gaĺı, Fabio Ghironi, DaleHenderson, Ben Hunt, Steve Kamin, Doug Laxton, Andrew Levin, StephenMurchison, Paolo Pesenti, Alessandro Rebucci, Trevor Reeve, Stephanie Schmitt-Grohe, Christopher Sims (discussant), Ralph Tryon, Martin Uribe, and partici-pants in workshops at the Bank of Canada, Duke University, the Bank of Finland,the Federal Reserve Board, the Federal Reserve Bank of Dallas, GeorgetownUniversity, and the University of Montreal. The views expressed in this paperare solely the responsibility of the authors and should not be interpreted as re-flecting the views of the Board of Governors of the Federal Reserve System orof any other person associated with the Federal Reserve System. Correspond-ing author: Erceg, telephone 202-452-2575, fax 202-872-4926. E-mail addresses:[email protected], [email protected], [email protected].

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2 International Journal of Central Banking March 2006

a heavy emphasis on fitting short-run properties of the data, aca-demic models increasingly heeded the methodological imperative ofthe RBC literature that required optimizing-agent foundations andmodel-consistent expectations. But the focus of the latter on co-herent theoretical underpinnings came at the expense of empiricalrealism.

In recent years, there has been a surge of interest in develop-ing optimization-based models that are more suited to fitting thedata. Consistent with this more empirical orientation, “state-of-the-art” stochastic dynamic general equilibrium (SDGE) models haveevolved to include a large array of nominal and real rigidities. Impor-tant work by Christiano, Eichenbaum, and Evans (2005) showed thattheir optimization-based model, which includes both sticky nominalwages and various types of adjustment costs in the expenditure com-ponents, is quite successful in accounting for the estimated effects ofa monetary policy shock. Smets and Wouters (2003) demonstratedthat the forecasting ability of a similar model appears comparableto that of an unconstrained Bayesian vector autoregression.

A salient motivation of this recent research has been to enhancethe latitude of the SDGE models to contribute to policy analysis.In this vein, a number of central banks and other institutions suchas the International Monetary Fund are in the process of developingmicrofounded models that can provide quantitative input into thepolicy process.1 But while recent empirical work validating certainfeatures of the microfounded models is encouraging, it remains anopen question whether these models can yield plausible implicationsacross the rather broad spectrum of shocks considered routinely inpolicy work.

In this paper, we address this question using a new multicoun-try open economy SDGE model (SIGMA) that we have developedfor quantitative policy analysis. Our new model has its antecedentsin the seminal open economy modeling framework of Obstfeld andRogoff (1995). However, it includes many of the nominal and real fric-tions that have been identified as empirically important in the closedeconomy models of Christiano, Eichenbaum, and Evans (2005) andSmets and Wouters (2003), such as habit persistence in consumption

1The IMF’s new SDGE model “GEM” is described by Laxton and Pesenti(2003). The Bank of England’s new model “BEQM” is described by Harrison etal. (2005).

Vol. 2 No. 1 SIGMA: A New Open Economy Model 3

and adjustment costs in investment. Moreover, it incorporates analo-gous frictions relevant in an open economy framework, including bothlocal currency pricing (e.g., Betts and Devereux [1996] or Devereuxand Engel [2002]) and costs of adjusting trade flows.

Our approach consists of comparing the implications of SIGMAwith those of the FRB/Global model, a large-scale econometricmodel used extensively in policy analysis at the Federal ReserveBoard.2 Our comparisons involve examining the impulse responsefunctions to a number of shocks often considered in policy work.These include domestic monetary and fiscal shocks, a taste shock tohome (U.S.) consumption demand, a shock to the risk premium inthe uncovered interest parity equation, and alternative shocks to for-eign demand. The large-scale models form an important benchmarkfor evaluating the new SDGE models both because they provide areasonably accurate reduced-form characterization of the data andbecause they embed the priors of policymakers who have appliedthese models to policy questions for several decades.3 While we donot exclude an eventual departure from the responses of the large-scale econometric models if the data provide sufficient grounds fordoing so, it seems crucial that the new SDGE models not a priorirule out such responses due to arbitrary structural restrictions in thetheoretical framework.

2While the FRB/Global model essentially has neoclassical properties in thelong run, the behavioral equations are formulated to allow considerable flexibilityin accounting for the short-run properties of the data. This model assumes thatexpectations are formed adaptively, i.e., expectations are derived from small-scale vector autoregressions. In our analysis, we focus on the implications forthe U.S. block of the model, which consists of about 80 estimated behavioralequations and 300 identities; the foreign sector consists of 29 country blocksand roughly 4,000 equations. Brayton et al. (1997) provide a description of themodel’s basic structure. The U.S. block of the FRB/Global model, augmentedwith a small external sector, can also be run as an independent model (FRB/US)under either adaptive or rational expectations. See Brayton and Tinsley (1996)for an overview; Kiley (2001) describes the specification and estimation of thebusiness investment sector; and Elmendorf and Reifschneider (2002) provide anapplication to fiscal policy.

3We also provide some comparisons to estimates from structural vector autore-gressions (SVARs). However, comparisons with FRB/Global have the advantagethat we can consider a larger set of shocks than typically analyzed in the SVARliterature. Moreover, estimates from the SVAR literature often show considerabledivergence, making it difficult to gauge the appropriate benchmark.


Our model comparisons indicate that the short-run responses ofSIGMA are qualitatively similar to FRB/Global for a large arrayof macroeconomic variables, including output, prices, interest rates,real exchange rates, and the trade balance. We highlight two featuresof our framework that are instrumental in giving SIGMA greaterflexibility to generate responses that are more closely aligned withFRB/Global quantitatively, including more persistent responses ofboth real and nominal variables. These features include “informa-tion frictions,” which posit agents as having incomplete informationabout the persistence of shocks, and non-Ricardian households.

With information frictions, agents use the Kalman filter to esti-mate the nature of shocks affecting the economy and to make pro-jections. This simple learning mechanism typically implies gradualresponses to underlying shocks that are similar to adaptive expec-tations models, while retaining the appealing property of model-consistent expectations. As emphasized by Erceg and Levin (2003),it implies that model dynamics may depend crucially on the credi-bility and transparency of policy changes.

Our model breaks Ricardian equivalence by assuming that somefraction of households simply consume their current after-tax dispos-able income.4 Accordingly, the short-run fiscal multiplier associatedwith temporary increases in government spending exceeds unity andthe response of private consumption is positive. The inclusion of bothnon-Ricardian agents and of information frictions in our model canaccount for a highly persistent fiscal multiplier similar to that inFRB/Global.5

But our comparisons also reveal some noticeable quantitative dis-parities between the SIGMA and FRB/Global models. First, im-port prices adjust much more quickly and completely to exchangerate changes in SIGMA, with full exchange rate passthrough after afew quarters. Second, SIGMA tends to imply substantially greatervolatility in the expenditure components of GDP than FRB/Global.Finally, SIGMA tends to imply smaller and less persistent spillovereffects from foreign shocks to the domestic economy (though the

4Our approach is similar to that of Gaĺı, López-Salido, and Vallés (2004) andMankiw (2000).

5The highly persistent fiscal multiplier and positive consumption response arein line with the empirical findings of Fatás and Mihov (2001) and of Blanchardand Perotti (2002).


disparity is small if the increase in foreign demand is investmentdriven).

We argue that these differences are particularly significant inso-far as they stem from features of our SDGE framework that appearrobust to reasonable departures from our benchmark calibration;specifically, they reflect certain restrictive theoretical constraintsthat are likely to hold in a broad class of current open economySDGE models. For instance, high exchange rate passthrough aftera few quarters is likely to characterize any model in which the de-sired price markup is either fixed (as in SIGMA) or exhibits onlymodest variation. In turn, the large volatility in expenditure com-ponents is partly attributable to high passthrough, and to a markedsensitivity of private absorption to very persistent changes in realinterest rates. Finally, insofar as SIGMA relies on trade linkages toaccount for spillover effects from foreign demand disturbances—afeature common to most open economy SDGE models—it may un-derstate the importance of spillovers arising through other channels,including financial linkages.

Notwithstanding some limitations, the SDGE framework pos-sesses some key advantages over existing econometric models as atool for policy analysis. The last section of our paper focuses onsome of these advantages in the context of longer-term simulations ofSIGMA, including simulations of cuts in distortionary tax rates, andof a productivity acceleration. One advantage of the SDGE frame-work is that it facilitates assessing how structural features of theeconomy affect its responses to shocks. For example, it is often of in-terest to consider how the effects of a tax cut depend on the elasticityof household labor supply, or on the extent to which households areRicardian in their consumption behavior. Such experiments are moredifficult to design in typical large-scale econometric models in whichthere may exist no clear linkage between structural features andreduced-form parameters. Another major advantage of the SDGEframework for policy questions is that it fully articulates the chan-nels through which the economy returns to its balanced growth pathfollowing initial “imbalances.” For instance, in the case of the pro-ductivity acceleration, we illustrate the economic forces that inducethe trade balance to eventually move into surplus following an initialtrade deficit, and discuss how different initial perceptions about theunderlying shock influence adjustment dynamics.


The remainder of this paper is organized as follows. Section 1presents our basic open economy model. The calibration is discussedin section 2. Section 3 compares short-run impulse response func-tions in SIGMA and FRB/Global for an array of shocks frequentlyconsidered in policy analysis. Section 4 examines long-run simula-tions in SIGMA under alternative calibrations. Section 5 concludesand provides a discussion of directions for future research.

1. The Model

Our model consists of two countries that may differ in size, but areotherwise isomorphic.6 Hence, our exposition below focuses on the“home” country. Each country in effect produces a single domesticoutput good, although we adopt a standard monopolistically com-petitive framework to rationalize stickiness in the aggregate pricelevel. While household utility depends on consumption of both thedomestic output good and imported goods, it is convenient to as-sume that a competitive distribution sector purchases both inputsand simply resells a final consumption good to households. Simi-larly, we assume that competitive distributors combine the domesticoutput good with imports to produce a final investment good.

To introduce non-Ricardian consumption behavior, we assumethat there are two types of households. One type of household max-imizes welfare subject to an intertemporal budget constraint. Thesehouseholds own the entire capital stock, accumulate capital subjectto adjustment costs, and exhibit habit persistence in their consump-tion decisions. They are also regarded as monopolistic competitorsin the labor market to account for aggregate wage stickiness. Theother type of household (the “hand-to-mouth” household) simplyconsumes its entire after-tax disposable income.

6For expositional purposes, we focus on a two-country variant of SIGMA.However, in actual policy analysis we typically use a variant with seven countryblocks that includes the United States, the euro area, Japan, Canada, Mexico,Developing Asia, and a rest-of-the-world sector. Moreover, the expanded modelincorporates features designed to account for the effects of oil shocks, whichinclude allowing oil to enter both the consumption bundle of households, andthe production function of firms (see Guerrieri 2005). For the shocks that weanalyze in this study, the effects on the home country (i.e., the United States)are quantitatively very similar in the two-country model discussed below as inthe larger policy model.


1.1 Firms and Price Setting

Production of Domestic Intermediate Goods. There is acontinuum of differentiated intermediate goods (indexed by i ∈ [0, 1])in the home country, each of which is produced by a single monop-olistically competitive firm. As in Betts and Devereux (1996), in-termediate goods firms charge different prices at home and abroad(i.e., they practice local currency pricing). In the home market, firmi faces a demand function that varies inversely with its output pricePDt(i) and directly with aggregate demand at home YDt:

YDt(i) =[PDt(i)PDt

]−(1+θ p )θ p

YDt, (1)

where θp > 0, and PDt is an aggregate price index defined below.Similarly, in the foreign market, firm i faces the demand function

Xt(i) =[P ∗Mt(i)P ∗Mt

]−(1+θ p )θ p

M∗t , (2)

where Xt(i) denotes the foreign quantity demanded of home goodi, P ∗Mt(i) denotes the price that firm i sets in the foreign market(denominated in foreign currency), P ∗Mt is the foreign import priceindex, and M∗t is aggregate foreign imports (we use an asterisk todenote foreign variables).

Each producer utilizes capital services Kt (i) and a labor indexLt (i) (defined below) to produce its respective output good. Theproduction function is assumed to have a constant elasticity of sub-stitution (CES) form:

Yt(i) =(ω

ρ1+ρ

K Kt(i)1

1+ρ + ωLρ

1+ρ (ZtLt(i))1

1+ρ)1+ρ

. (3)

The production function exhibits constant-returns-to-scale in bothinputs, and technological progress Zt is given by

Zt = exp(gzt + zt), (4)

where zt is a country-specific shock to the level of technology and gz ,the deterministic rate of technological growth, is assumed to be thesame in both countries. Firms face perfectly competitive factor mar-kets for hiring capital and labor. Thus, each firm chooses Kt (i) and


Lt (i), taking as given both the rental price of capital RKt and theaggregate wage index Wt (defined below). Firms can costlessly ad-just either factor of production. Thus, the standard static first-orderconditions for cost minimization imply that all firms have identicalmarginal cost per unit of output, MCt.

We assume that the home and foreign prices of the intermediategoods are determined by Calvo-style staggered contracts (see Calvo1983). In each period, a firm faces a constant probability, 1 − ξp,of being able to reoptimize its price at home (PDt(i)) and 1 − ξp,xprobability of being able to reoptimize its price abroad (P ∗Mt(i)).These probabilities are assumed to be independent across firms, time,and countries. If a firm is not allowed to optimize its prices, we fol-low Christiano, Eichenbaum, and Evans (2005) and assume the firmmust reset its home price based on lagged aggregate inflation (i.e.,PDt(i) = πt−1PDt−1(i), where πt = PDt/PDt−1).7 In foreign mar-kets, if a firm cannot reoptimize its price, the price is changed ac-cording to an analogous rule (i.e., P ∗Mt(i) = π

∗Mt−1P

∗Mt−1(i), where

π∗Mt = P∗Mt/P

∗Mt−1). This form of lagged indexation is a mech-

anism for introducing inflation inertia into the key price-settingequations.

When firm i is allowed to reoptimize its price in the domesticmarket in period t, the firm maximizes

Ẽt

∞∑j=0

ξjpψt,t+j [VDt+jPDt (i)YDt+j (i) − MCt+jYDt+j (i)] . (5)

The operator Ẽt represents the conditional expectation based on theinformation available to agents at period t. The firm discounts profitsreceived at date t + j by the state-contingent discount factor ψt,t+j ;for notational simplicity, we have suppressed all of the state indices.8

7In alternative calibrations of SIGMA, we also consider the specification usedby Yun (1996) and Erceg, Henderson, and Levin (2000) where PD t (i) = πPD t−1(i)so that VD t+j = π

j in the profit functional defined below. For this alternativecalibration, prices are updated according to PM t (i) = π

∗PM t−1(i) in foreignmarkets.

8We define ξt,t+j to be the price in period t of a claim that pays one dollar ifthe specified state occurs in period t+ j (see the household problem below); thenthe corresponding element of ψt,t+j equals ξt,t+j divided by the probability thatthe specified state will occur.


Also, VDt+j is defined by

VDt+j =j∏

h=1

πt+h−1. (6)

The first-order condition for setting the contract price of good i inthe home market is

Ẽt

∞∑j=0

ψt,t+jξjp

(VDt+jPDt (i)

(1 + θp)− MCt+j

)YDt+j (i) = 0. (7)

Defining a similar profit functional to equation (5) for a firm’s opti-mal choice of its contract price in the foreign market at date t, theassociated first-order condition is

Ẽt

∞∑j=0

ψt,t+jξjp,x

(et+jV

∗Mt+jP

∗Mt(i)

(1 + θp)− MCt+j

)Xt+j (i) = 0. (8)

In equation (8), et is the price of a unit of foreign currency in terms ofthe home currency (so that a rise in et corresponds to a depreciationof the home currency), and V ∗Mt+j is defined as

V ∗Mt+j =j∏

h=1

π∗Mt+h−1. (9)

Production of the Domestic Output Index. Because house-holds have identical Dixit-Stiglitz preferences, it is convenient toassume that a representative aggregator combines the differentiatedintermediate products into a composite home-produced good YDt:

YDt =[∫ 1

0YDt (i)

11+θ p di

]1+θp. (10)

The aggregator chooses the bundle of goods that minimizes the costof producing YDt, taking the price PDt (i) of each intermediate goodYDt(i) as given. The aggregator sells units of each sectoral outputindex at its unit cost PDt:

PDt =[∫ 1

0PDt (i)

−1θ p di

]−θp. (11)


We also assume a representative aggregator in the foreign economywho combines the differentiated home products Xt(i) into a singleindex for foreign imports:

M∗t =[∫ 1

0Xt (i)

11+θ p di

]1+θp, (12)

and sells M∗t at price P∗Mt:

P ∗Mt =[∫ 1

0P ∗Mt (i)

−1θ p di

]−θp. (13)

Production of Consumption and Investment Goods. Finalconsumption goods are produced by a representative “consumptiongoods distributor.” This firm combines purchases of domesticallyproduced goods with imported goods to produce a final consumptiongood (Ct) according to a constant-returns-to-scale CES productionfunction:

Ct =(

ωρC

1+ρCC C

11+ρCDt + (1 − ωC )

ρC1+ρC (ϕCtMCt)

11+ρC

)1+ρC, (14)

where CDt denotes the consumption goods distributor’s demand forthe index of domestically produced goods, MCt denotes the distrib-utor’s demand for the index of foreign-produced goods, and ϕCtreflects costs of adjusting consumption imports. The form of theproduction function mirrors the preferences of households over con-sumption of domestically produced goods and imports. Accordingly,the quasi-share parameter ωC may be interpreted as determininghousehold preferences for home relative to foreign goods, or equiva-lently, the degree of home bias in household consumption expendi-ture. Finally, the adjustment cost term ϕCt is assumed to take thequadratic form

ϕCt =

1 − ϕMC2

MC tCD tMC t−1CD t−1

− 1

2 . (15)This specification implies that it is costly to change the proportionof domestic and foreign goods in the aggregate consumption bundle,


even though the level of imports may jump costlessly in responseto changes in overall consumption demand. It aims to capture theintuitively appealing notion that households may have limited abil-ity in the short run to vary the mix of domestic relative to foreigngoods in producing consumption services, even if longer-run substitu-tion possibilities are more favorable. From an empirical perspective,our specification is consistent with evidence which suggests that im-ports adjust slowly in response to relative price changes, but respondrapidly to changes in real activity, e.g., McDaniel and Balistreri(2003).

Given the presence of adjustment costs, the representative con-sumption goods distributor chooses (a contingency plan for) CDtand MCt to minimize its discounted expected costs of producing theaggregate consumption good:

minCD t+k ,MC t+k

Ẽt

∞∑k=0

ψt,t+k

{(PDt+kCDt+k + PMt+kMCt+k)

+PCt+k

[Ct+k −

(ω

ρC1+ρCC C

11+ρCDt+k + (1 − ωC )

ρC1+ρC

(ϕCt+kMCt+k)1

1+ρC

)1+ρC ]}. (16)

The distributor sells the final consumption good to households at aprice PCt, which may be interpreted as the consumption price index(or equivalently, as the shadow cost of producing an additional unitof the consumption good).

We model the production of final investment goods in an anal-ogous manner. Thus, the representative “investment goods distrib-utor” produces a final investment good by combining its purchasesof domestically produced goods with purchases of foreign-producedgoods according to a constant-returns-to-scale CES productionfunction:

It =(

ωρI

1+ρII I

11+ρIDt + (1 − ωI )

ρI1+ρI (ϕItMIt)

11+ρI

)1+ρI, (17)

where IDt denotes the investment goods distributor’s demand for theindex of domestically produced goods, MIt denotes the distributor’s


demand for the index of foreign goods, and ϕIt reflects costs of ad-justing imports of investment goods. As in the case of consumptiongoods, the quasi-share parameter ωI may be interpreted as deter-mining the degree of home bias in the production of final investmentgoods. The adjustment cost ϕIt takes a form that is analogous tothe adjustment cost specification for imported consumption goods,so that

ϕIt =

1 − ϕMI2

MI tID tMI t−1ID t−1

− 1

2 . (18)Investment goods distributors solve an intertemporal cost mini-

mization problem isomorphic to that of consumption goods distrib-utors. Each distributor sells the final investment good to householdsat a price PIt, which may be interpreted as the investment priceindex. This price may differ from the price index of the consump-tion good PCt because of differences in import composition, even inthe absence of the adjustment costs for consumption and investmentimports.

1.2 Households and Wage Setting

We assume a continuum of monopolistically competitive households(indexed on the unit interval), each of which supplies a differenti-ated labor service to the intermediate goods-producing sector (theonly producers demanding labor services in our framework). It isconvenient to assume that a representative labor aggregator (or “em-ployment agency”) combines households’ labor hours in the sameproportions as firms would choose. Thus, the aggregator’s demandfor each household’s labor is equal to the sum of firms’ demands.The aggregate labor index Lt has the Dixit-Stiglitz form

Lt =[∫ 1

0(ζtNt (h))

11+θw dh

]1+θw, (19)

where θw > 0 and Nt(h) is hours worked by a typical member ofhousehold h. Also, ζt is the size of a household of type h and evolvesaccording to ζt = gnζt−1 (effectively, ζt and gn determine the sizeand growth rate of the population). The aggregator minimizes thecost of producing a given amount of the aggregate labor index, taking


each household’s wage rate Wt (h) as given, and then sells units ofthe labor index to the production sector at their unit cost Wt:

Wt =[∫ 1

0Wt (h)

−1θw dh

]−θw. (20)

It is natural to interpret Wt as the aggregate wage index. The aggre-gator’s demand for the labor services of a typical member of house-hold h is given by

Nt (h) =[Wt (h)

Wt

]− 1+θwθw

Lt/ζt. (21)

We assume that there are two types of households: (i) householdsthat make intertemporal consumption, labor supply, and capital ac-cumulation decisions in a forward-looking manner by maximizingutility subject to an intertemporal budget constraint (FL households,for “forward-looking” households) and (ii) the remainder that sim-ply consume their after-tax disposable income (HM households, for“hand-to-mouth” households). The HM households receive no capi-tal rental income or profits, and choose to set their wage to be theaverage wage of optimizing households. Given that households ofeach type grow at the same rate, the share of each type of householdin the population is fixed. We denote the share of FL householdsby ς and the share of HM households by 1 − ς.

We consider first the problem faced by FL households. Theutility functional for an optimizing representative member of house-hold h is

Ẽt

∞∑j=0

βj

11 − σ(

Ct+j(h) −κ

ς

COt+j−1ζt+j−1

− νct

)1−σ

+χ0Z

1−σt+j

1 − χ (1 − Nt+j(h))1−χ +

µ01 − µ

(MBt+j+1(h)

PCt+j

)1−µ}, (22)

where the discount factor β satisfies 0 < β < 1. As in Smetsand Wouters (2003), we allow for the possibility of external habits,where each household member cares about its consumption relativeto lagged aggregate consumption per capita of optimizing agents,


COt−1ςζt−1

. The period utility function depends on each member’s cur-rent leisure 1−Nt(h), his or her end-of-period real money balances,MBt+1(h)

PC t, and a preference shock, νct. We allow for preferences over

leisure to shift with the level of technology so that the model is con-sistent with balanced growth, even if the subutility function overconsumption is not logarithmic.9

Household h faces a flow budget constraint in period t whichstates that its combined expenditure on goods and on the net accu-mulation of financial assets must equal its disposable income:

PCtCt(h) + PItIt(h) + MBt+1(h) − g−1n MBt(h) +∫s ξt,t+1BDt+1(h)

−g−1n BDt(h) + PBtBGt+1 − g−1n BGt +et P ∗B t BF t+1(h)

φbt− g−1n etBF t(h)

= (1 − τNt)Wt(h)Nt(h) + Γt(h) + TRt(h) − Tt(h)+(1 − τKt)g−1n RKtKt(h) + PItτKtδg−1n Kt(h) − PDtφIt(h). (23)

The presence of the population growth parameter gn in the house-hold’s budget constraint reflects that equation (23) is expressed inper capita terms as well as the assumption that new household mem-bers are born without any initial holdings of bonds, capital, or money.Final consumption goods are purchased at a price PCt, and final in-vestment goods at a price PIt. Investment in physical capital aug-ments the per capita capital stock Kt+1(h) according to a lineartransition law of the form

Kt+1 (h) = (1 − δ)g−1n Kt(h) + It(h), (24)

where δ is the depreciation rate of capital.Financial asset accumulation of a typical member of FL house-

hold h consists of increases in nominal money holdings (MBt+1 (h)−g−1n MBt (h)) and the net acquisition of bonds. We assume thatagents within a country can engage in frictionless trading of acomplete set of contingent claims, while trade in international as-sets is restricted to a non-state-contingent nominal bond. The

9This statement is only strictly true in the absence of permanent country-specific technology shocks. In this case, a permanent increase in technology inthe home country that does not occur abroad will be associated with a permanentdeterioration in the home country’s terms of trade that moves the home economyoff its balanced growth path.


term PBtBGt+1 − g−1n BGt represents each household member’s netpurchases of domestic government bonds, while

∫s ξt,t+1BDt+1(h) −

g−1n BDt(h) is net purchases of state-contingent domestic bonds. Wedenote ξt,t+1 as the price of an asset that will pay one unit of do-mestic currency in a particular state of nature at date t + 1, whileBDt+1 (h) represents the quantity of such claims purchased by a typ-ical member of household h at time t. Thus, the gross outlay onnew state-contingent domestic claims is given by integrating over allstates at time t + 1, while BDt (h) indicates the value of the house-hold’s existing claims (on a per capita basis) given the realized stateof nature.

In equation (23), BF t+1(h) represents the quantity of a non-state-contingent bond purchased by a typical member of household h attime t that pays one unit of foreign currency in the subsequent pe-riod, P ∗Bt is the foreign currency price of the bond, and et is theexchange rate expressed in units of home currency per unit of for-eign currency. We follow Turnovsky (1985) and assume there is anintermediation cost φbt paid by households in the home country forpurchases of foreign bonds, which ensures that net foreign assets arestationary in the model.10 More specifically, the intermediation costsdepend on the ratio of economy-wide holdings of net foreign assetsto nominal GDP, PtYt, and are given by

φbt = exp(−φb

(etBF t+1

PtYt

)+ νbt

). (25)

In the above, νbt is a mean-zero stochastic process, which we interpretas a risk-premium shock or shock to the uncovered interest-rate par-ity condition. Abstracting from this shock, if the home economy hasan overall net lender position internationally, then a household willearn a lower return on any holdings of foreign bonds. By contrast, ifthe economy has a net debtor position, a household will pay a higherreturn on its foreign liabilities.

Each member of FL household h earns after-tax labor income,(1−τNt)Wt (h) Nt (h), where τNt is a stochastic tax on labor income.The household leases capital to firms at the after-tax rental rate

10This intermediation cost is asymmetric, as foreign households do not facethese costs; rather, they collect profits on the monopoly rents associated withthese intermediation costs.


(1 − τKt)RKt, where τKt is a stochastic tax on capital income. Thehousehold receives a depreciation write-off of PItτKtδ per unit ofcapital. Each member also receives an aliquot share Γt (h) of theprofits of all firms and a lump-sum government transfer, TRt (h),and pays a lump-sum tax Tt(h). Following Christiano, Eichenbaum,and Evans (2005), we assume that it is costly to change the level ofgross investment from the previous period, so that the accelerationin the capital stock is penalized:

φIt(h) =12φI

(It(h) − gzgnIt−1(h))2

It−1(h). (26)

In every period t, each member of FL household h maximizes theutility functional (22) with respect to its consumption, investment,(end-of-period) capital stock, money balances, holdings of contingentclaims, and holdings of foreign bonds, subject to its labor demandfunction (21), budget constraint (23), and transition equation forcapital (24). In doing so, a household takes as given prices, taxesand transfers, and aggregate quantities such as lagged aggregate con-sumption and the aggregate net foreign asset position.

Forward-looking (FL) households set nominal wages in staggeredcontracts that are analogous to the price contracts described above.In particular, with probability 1−ξw , each member of a household isallowed to reoptimize its wage contract. If a household is not allowedto optimize its wage rate, we assume each household member resetsits wage according to

Wt(h) = ωt−1Wt−1(h), (27)

where ωt = Wt/Wt−1 and in steady state ω = πgz .11 Each memberof household h chooses the value of Wt(h) to maximize its utilityfunctional (22), yielding the following first-order condition:

Ẽt∑∞

j=0 βjξjw{ 1−τ N t(1+θw )

Λt+jPt+j

Vwt+jWt(h)

−χ0t+jZ1−σt+j (1 − Nt+j(h))−χ}Nt+j(h) = 0, (28)

11In alternative specifications, we also consider Wt (h) = ωWt−1(h).


where Λt is the marginal value of a unit of consumption, and Vwt+jis defined as

Vwt+j =j∏

h=1

ωt+h−1. (29)

Roughly speaking, equation (28) says that the household chooses itscontract wage to equate the present discounted value of working anadditional unit of time to the discounted marginal cost.

Finally, we consider the determination of consumption and laborsupply of the hand-to-mouth (HM) households. A typical memberof an HM household simply equates his or her nominal consump-tion spending to his or her current after-tax disposable income,which consists of labor income plus net lump-sum transfers fromthe government:

PCtCt (h) = (1 − τNt)Wt (h)Nt (h) + TRt(h) − Tt (h) . (30)

The HM households set their wage to be the average wageof the forward-looking households. Since HM households face thesame labor demand schedule as the forward-looking households, eachHM household works the same number of hours as the average forforward-looking households.

1.3 Monetary Policy

We assume that the central bank follows an interest rate reactionfunction similar in form to the historical rule estimated by Or-phanides and Wieland (1998) over the Volcker-Greenspan period.Thus, the short-term nominal interest rate is adjusted so that theex post real interest rate rises when inflation exceeds its constanttarget value, or when output growth rises above some target value.With some allowance for interest rate smoothing, monetary policy isdescribed by the following interest rate reaction function:

it = γiit−1 + r + πt + γπ(π(4)t − π) + γy(yt − yt−4 − gy) + �it. (31)

In the above, it is the annualized nominal interest rate, π(4)t is

the four-quarter inflation rate of the GDP deflator (i.e., π(4)t =∑3j=0 πt−j), and r and π are the steady-state real interest rate and


the central bank’s constant inflation target (both expressed at an-nual rate). Also, yt − yt−4 is the four-quarter growth rate of output,and gy is its corresponding steady-state value.

1.4 Fiscal Policy

Some of the domestically produced good is purchased by the govern-ment. Government purchases (Gt) are assumed to have no direct ef-fect on the utility of a household.12 We also assume that governmentpurchases as a fraction of output, gt = PD t GtPt Yt , follow an exogenousstochastic process.

The government can issue debt BGt+1 to finance a deficit so thatits budget constraint is given by

PBtBGt+1 − BGt = PDtGt + TRt − Tt − τNtWtLt−(τKtRKt − δPIt)Kt−(MBt+1 − MBt). (32)

In equation (32), we have aggregated the capital stock, money andbond holdings, and transfers and taxes over all households so that, forexample, Tt = ζt

∫ 10 Tt(h)dh. As noted above, labor and capital taxes

are determined exogenously, while we assume that real transfers asa fraction of domestic output, trt = T RtPt Yt , evolve according to anexogenous stochastic process. Given that the central bank uses thenominal interest rate as its policy instrument, the level of seignioragerevenues is determined by nominal money demand.

Lump-sum taxes are adjusted in a manner that the governmentsatisfies an intertemporal solvency constraint, requiring that thepresent discounted value of the government debt stock tends towardzero in the long run. In particular, we assume that the real lump-sumtax rate, τ t = TtPt Yt , is determined according to the following reactionfunction:

τ t = ν0τ t−1 + ν1(bGt+1 − bG) + ν2(bGt+1 − bGt), (33)

12The model’s dynamics would be unchanged if we had assumed instead thatgovernment purchases entered separably in the utility function, although thewelfare consequences would be different.


where bGt+1 =BG t+1Pt Yt

and bG is the government’s target value for theratio of government debt to nominal output.13

1.5 Resource Constraint and Net Foreign Assets

The home economy’s aggregate resource constraint can be written as

YDt = CDt + IDt + Gt + φIt, (34)

and φIt is the adjustment cost on investment aggregated across allhouseholds. Total exports may be allocated to either the consump-tion or the investment sector abroad:

M∗t = M∗Ct + M

∗It. (35)

Finally, at the level of the individual firm,

Yt(i) = YDt(i) + Xt(i) ∀i. (36)

The evolution of net foreign assets can be expressed as

etP∗B,tBF,t+1

φbt= etBF,t + etP ∗MtM

∗t − PMtMt. (37)

This expression can be derived from the budget constraint of the FLhouseholds after imposing the government budget constraint, theconsumption rule of the HM households, the definition of firm prof-its, and the condition that domestic bonds (BDt+1) are in zero netsupply.14

Finally, we assume that the structure of the foreign economy (the“rest of the world”) is isomorphic to that of the home country.

13We found that the inclusion of the term (bG t+1 − bG t ) was instrumental inyielding a determinate rational expectations equilibrium over a large region ofthe parameter space.

14The derivation of the evolution of net foreign assets also requires thatPC tCt = PD tCD t + PM tMC t and PI t It = PD t ID t + PM tMI t . Given that theseconditions are satisfied even in the presence of adjustment costs for imports, theimport adjustment cost terms do not appear in equation (37).


1.6 Description of Shocks and the Optimal Filtering Problem

As discussed above, our model includes shocks to the level of homeproductivity (Zt), real government spending (gt), real transfers (trt),labor tax rate (τNt), capital tax rate (τKt), preferences (νct), andsimilar shocks to the foreign country (which may be regarded asalternative sources of foreign demand shocks from the perspective ofthe home country). In addition, it includes a shock to the financialintermediation technology (νbt), which can be interpreted as a riskpremium shock.

We assume that each shock in the model has two underlyingcomponents with different levels of persistence. Agents observe theshock, but they cannot observe the separate components. It is helpfulto illustrate the nature of the forecasting problem with reference to aparticular shock, since the treatment of the other shocks is basicallyisomorphic. Thus, focusing on the case of the productivity growthshock examined in section 4 below, and recalling from equation (4)that productivity growth can be written as ∆ log(Zt) = gz + ∆zt,our approach attributes changes in zt to two separate components:

∆zt = ∆zP t + ∆zT t. (38)

The first component, ∆zP t, is a highly persistent shock that shiftsthe “trend” level of productivity growth, and the second, ∆zT t, is atransient shock to productivity growth. The bivariate process deter-mining the evolution of each component may be represented as ∆zP t

∆zT t

= ρp 0

0 0

∆zP t−1∆zT t−1

+ εP t

εT t

, (39)where the persistence parameter ρp is assumed to be slightly belowunity, and the transient shock is assumed to be i.i.d. in the case ofthis particular shock (for other shocks, e.g., to government spending,we allow the transient component to be somewhat persistent, thoughmuch less persistent than the permanent shock). Thus, an innovationto the growth rate of the transient component has a permanent effecton the level of productivity, but no effect on the future growth rate.Moreover, the innovations εP t and εT t are mutually uncorrelatedwith variances vP and vT , respectively.


Agents observe the current level of productivity in the economy,and hence observe ∆zt, but cannot observe the growth rate of theunderlying components ∆zP t and ∆zT t. Thus, agents solve a signalextraction problem to forecast the future level of productivity. Giventhat agents know the underlying parameters of the bivariate processfor productivity growth, the Kalman filter can be used to obtain anoptimal solution.

The expected level of productivity at a future date K periodsahead depends only on the current level of productivity and on theexpected growth rate of the permanent component:

Ẽt log(Zt+K ) = Kgz + log(Zt) + Ẽt(zP t+k − zP t)

= Kgz + log(Zt) +K∑

J=1

Ẽt∆zP t+J . (40)

The Kalman filtering algorithm implies that the expected growthrate of the permanent component is updated according to

Ẽt∆zP t = ρpẼt−1∆zP t−1 + kgρp(∆zt − ρpẼt−1∆zP t−1). (41)

Thus, agents update their assessment of the persistent componentof the productivity growth process by the product of the forecasterror innovation and a constant coefficient. This coefficient, which isproportional to the scalar Kalman gain parameter kg, is an increasingfunction of the signal-to-noise ratio vPvT (the ratio of the variances ofthe persistent and transitory components of the productivity growthprocess).

2. Solution Method and Calibration

Because the levels of technology and the population are nonstation-ary, real variables (including output and the expenditure compo-nents of GDP) are also nonstationary. Accordingly, prior to solvingthe model, we scale real variables in the home and foreign countryby the common deterministic trends in technology and populationsize. Nominal variables are scaled to account both for growth in thecorresponding real variable and for the steady-state inflation rate.

We solve the model by log-linearizing the equations (specified interms of the transformed variables) around the steady state associ-ated with common growth rates of technology and population in the


two countries. To obtain the reduced-form solution of the model,we use the numerical algorithm of Anderson and Moore (1985),which provides an efficient implementation of the method proposedby Blanchard and Kahn (1980) (see also Anderson 1997).15

2.1 Calibration of Parameters

The model is calibrated at a quarterly frequency. Structural param-eters are set at identical values for each of the two countries, exceptfor the parameters determining population size (as discussed below).We assume that the discount factor β = 0.997 and the rate of tech-nological growth gz = 1.0037. These values are consistent with asteady-state annualized real interest rate r of about 3 percent.

The utility functional parameter σ is set equal to 2, while the pa-rameter determining the degree of habit persistence in consumptionκ = 0.8. We set χ = 10, implying a Frisch elasticity of labor supplyof 1/5, which is considerably lower than if preferences were logarith-mic in leisure, but well within the range of most empirical estimates.The utility parameter χ0 is set so that employment comprises one-third of the household’s time endowment, while the parameter µ0 onthe subutility function for real balances is set at an arbitrarily lowvalue (although given the separable specification, variation in realbalances has no impact on other variables). We choose ς = 0.5 sothat 50 percent of households are Ricardian FL agents.16

The depreciation rate of capital δ is set at 0.025 (consistent withan annual depreciation rate of 10 percent). We fix the price andwage markup parameters so that θp = θw = 0.20, similar to theestimated values obtained by Rotemberg and Woodford (1999) and

15The steady state around which we linearize depends on the relative levelof technology in each country, which we initialize to unity (so that per capitaincome in each country is identical in the steady state, though GDP may differacross countries due to population differences). We evaluated the robustness ofour solution procedure by using a nonlinear Newton-Raphson algorithm that doesnot rely on linearization around an initial steady state, and found that the resultswere nearly identical to those reported.

16Gaĺı, López-Salido, and Vallés (2004) found in a New Keynesian sticky pricemodel with HM agents that the equilibrium becomes indeterminate when theshare of HM agents is substantial. By contrast, our model produces a determinaterational expectations solution even for much higher values of the share of HMhouseholds than the 50 percent assumed in our baseline.


Amato and Laubach (2003).17 We set ξp and ξw to be consistent withfour-quarter contracts (subject to full indexation).18 The parameterξp,x is chosen to be consistent with two-quarter contracts.19 We setthe steady-state inflation rate π to yield an annual inflation rate of4 percent.

The parameter ρ in the CES production function of the inter-mediate goods producers is set to −2, implying an elasticity of sub-stitution between capital and labor of 1/2. Thus, capital and laborare less substitutable than the unitary elasticity case implied by theCobb-Douglas specification. The quasi-capital share parameter ωKis chosen to imply a steady-state investment to output ratio of 20percent. The private consumption to output ratio is 70 percent, whilegovernment consumption is 10 percent of steady-state output. We setthe cost of adjusting investment parameter φI = 3, slightly below thevalue used by Christiano, Eichenbaum, and Evans (2005).

The parameter ωC is chosen to match the estimated average shareof imports in total U.S. consumption of about 9 percent (based ondata from the U.S. Bureau of Economic Analysis), while the param-eter ωI is chosen to match the average share of imports in totalU.S. investment of about 38 percent. Given that trade is balancedin steady state, this parameterization implies an import or exportto GDP ratio for the home country (the United States) of about 13percent. We choose the initial population levels ζ0 and ζ

∗0 so that the

home country constitutes 25 percent of world output. This impliesan import (or export) share of output of the foreign country of about3 percent.

We assume that ρC = ρI = 2, consistent with a long-run priceelasticity of demand for imported consumption and investment goodsof 1.5. While this is higher than most empirical estimates using

17Rotemberg and Woodford (1999) found θp = 0.15, while Amato and Laubach(2003) obtained θp = 0.19 and θw = 0.13. Given our assumption that there isperfect capital mobility across firms within a country, the parameter θp onlyaffects steady-state relationships and does not otherwise appear in the dynamicequations of the log-linearized model.

18The inclusion of strategic complementarities along the lines suggested byKimball (1995) and Woodford (2003) would allow our model to generate similardynamic responses with shorter contract durations.

19The rapid adjustment of import prices is consistent with the evidence Campaand Goldberg (2004) derived from a panel of OECD countries, nothwithstandingtheir finding that long-run passthrough is generally well below 100 percent forOECD countries.


macrodata, we emphasize that the presence of adjustment coststranslates into a much lower relative price sensitivity in the shortto medium term. In particular, we set the adjustment cost param-eters ϕMC = ϕMI = 10, implying a price elasticity near unity afterfour quarters. We choose a small value (0.001) for the financial in-termediation cost φb, which is sufficient to ensure the model has aunique steady state.

We estimated the parameters of the monetary policy rule usingU.S. data from 1983:1–2003:4.20 Our estimates imply γπ = 0.6, γy =0.28, and γi = 0.8. For the tax rate reaction function, we chooseν0 = 1, ν1 = 0.1, ν2 = 0.001, and bG = 0. We set the steady-statecapital and labor tax rates equal to 0.3 and 0.2, respectively.

3. Comparisons of SIGMA and FRB/Global

We now turn to assessing the short-run properties of our SIGMAmodel, comparing the implications of SIGMA with those ofFRB/Global for an array of shocks often considered in policy analy-sis. To facilitate comparison across models, we specify the monetarypolicy rule in FRB/Global to be the same as in the benchmark ver-sion of SIGMA.

3.1 Loosening of Monetary Policy

We begin by examining the effects of a transient innovation to themonetary policy rule in SIGMA, i.e., a rise in �it in equation (31).The shock is scaled to induce an initial decline in the short-termnominal interest rate of about 75 basis points. As shown in figure 1,this policy shock raises output by slightly less than 0.2 percent aftertwo to three quarters. The response of investment is roughly threetimes larger than the response of consumption, reflecting the higherinterest sensitivity of the former. The fall in domestic real interestrates drives a modest depreciation of the real exchange rate (a risein the figure), which pushes up import prices (not shown). The com-bination of a positive output gap and higher import prices causesconsumer price inflation to rise.

20We estimated the rule using instrumental variables with lags of inflation andoutput growth as instruments.


Figure 1. Monetary Policy Loosening


While the figure shows that the qualitative effects of the shockare similar in FRB/Global, the peak effects on real GDP and the ex-penditure components are noticeably larger in FRB/Global than inSIGMA. This primarily reflects that long-term real interest rates(which are determined by a small-scale vector autoregression inFRB/Global) drop much more sharply and persistently than inSIGMA (in which long-term real rates are effectively determined bythe expectations hypothesis). Given this disparity in the long-termreal interest rate responses, it is useful to control for differences in thesimulation results that are attributable to alternative term structureequations. Accordingly, the dash-dotted line in the figure shows analternative calibration of the shock in SIGMA that implies a responseof the five-year real interest rate that comes very close to matchingthat in FRB/Global (this calibration assumes that the persistenceof the monetary policy error �it in the interest rate reaction functionis 0.6 rather than 0 as in our benchmark). In this case, the peakresponses of GDP and the real expenditure components are veryclose across the two models; the notable difference is in the inflationresponse, which is much larger in SIGMA.

These results indicate that the responses of the two models toa monetary policy shock that has similar effects on long-term realinterest rates is quite similar across the two models (at least for realvariables). However, we caution that the similarity in responses ap-pears sensitive to the persistence of the underlying shock. Thus, theresults should not be interpreted more broadly as indicating thatthe interest sensitivity of aggregate demand is similar across the twomodels, irrespective of persistence of the shock’s effect on real inter-est rates. As we show below, SIGMA appears to exhibit a somewhatgreater interest elasticity to shocks that exert highly persistent ef-fects on long-term real interest rates.21

21The results from our SIGMA model also appear broadly consistent with theimplications of the empirical vector autoregression (VAR) literature that esti-mates responses to a monetary policy innovation. For example, the results ofChristiano, Eichenbaum, and Evans (2005) indicate that a monetary policy in-novation that reduces the nominal interest rate by about 75 basis points inducesoutput to rise about 0.4 percent, which is very close to our model for the cali-bration in which the monetary policy error follows an AR(1) with a persistenceparameter of 0.6. However, direct comparisons of the quantitative effects aresomewhat difficult because model results are fairly sensitive to the form of themonetary policy rule, which differs from that imposed in the VAR.


3.2 Rise in Government Spending

Figure 2 shows the effects of an exogenous rise in the U.S. governmentspending share of GDP of 1 percentage point relative to baseline.The shock is highly persistent (ρg = 0.975), so that the governmentspending share remains about 0.6 percentage point above baselineafter five years.

The rise in government spending induces an immediate expan-sion of output. The government spending multiplier exceeds unityin the impact period due to the sharp rise in consumption of theHM households (as implied by the dotted line, the impact multiplierwould be below unity if all households were optimizers, reflectinga sharp immediate fall in aggregate consumption). However, risingreal interest rates quickly crowd out private investment and the con-sumption of the interest-sensitive optimizing households, which isdepressed further due to the negative wealth effect of higher govern-ment spending. In fact, the fall in consumption for the optimizinghouseholds is exacerbated by the presence of the HM households,as real interest rates rise even more than would occur if all house-holds were optimizers. Thus, overall private consumption falls belowbaseline after only a few quarters, and most of the output expansionis reversed. The small but more persistent component of the outputincrease reflects a rise in labor supply that is induced by the negativewealth effect.

Figure 2 also shows impulse responses derived from the FRB/Global model for a similar-sized rise in government spending. Clearly,the output response is noticeably more persistent in FRB/Global, asoutput remains nearly 0.4 percent above baseline three years afterthe shock occurs. The greater persistence of the output responsereflects that the crowding out of private investment and consump-tion spending occurs much more gradually. This partly reflects thatnegative wealth effects play a less prominent role in depressing con-sumption in FRB/Global. In addition, private absorption (especiallyinvestment) is considerably less sensitive to persistent changes inthe long-term real interest rate in FRB/Global, especially in theshort run. Thus, while the responses in SIGMA and FRB/Global aresimilar qualitatively, the responses in the latter tend to exhibit no-ticeably greater persistence. The empirical vector autoregression evi-dence of Blanchard and Perotti (2002) also suggests that government


Figure 2. Rise in Government Spending


spending shocks exert highly persistent effects on output and onthe expenditure components that appear more consistent with theFRB/Global responses.

A plausible channel for inducing greater persistence in the im-pulse responses of the SIGMA model is to allow for imperfect infor-mation, as also shown in the figure (dash-dotted lines).22 Becauseagents initially perceive that the increase in government spendingis temporary under imperfect information, there is a larger increasein GDP in this case than in the benchmark. Furthermore, the risein GDP is more persistent, as agents only slowly update their be-liefs about the persistence of the shock and are continually surprisedby the higher-than-expected levels of government spending. Givenboth a less-pronounced rise in real longer-term interest rates and asmaller negative wealth effect (since agents think the governmentspending rise will be temporary), there is much less crowding out ofconsumption and investment spending—with consumption even re-maining above baseline for a sustained duration. Thus, the inclusionof both non-Ricardian agents and information frictions would seemto provide a tractable channel for increasing the flexibility of SDGEmodels to account for responses that are similar to large-scale pol-icy models, and to encompass the range of responses derived fromempirical vector autoregressions.

3.3 Rise in Home Consumption Demand

Figure 3 assesses the effects of a taste shock νct to equation (22)that raises the marginal utility of consumption. This shock may beregarded as tantamount to the “autonomous shift in consumptiondemand” that is often considered in policy simulations. The shockis scaled so that it induces private consumption to rise by 1 percentabove baseline at peak impact, and has a persistence of 0.975.

The taste shock exerts a highly persistent positive effect on realinterest rates, accounting for the immediate rise in the five-year realinterest rate shown in the figure. As a result, the stimulative effectsof the rise in consumption demand on output are partly offset by a

22The temporary shock follows an AR(1) with a persistence parameter of 0.5.The innovation variances imply a Kalman gain parameter on the permanentcomponent of 0.07. We use a similar calibration for the home consumption tasteshock and foreign investment demand shocks considered below.


Figure 3. Rise in Home Consumption Demand


contraction in investment demand and by a reduction in real net ex-ports. The latter occurs because higher real interest rates generate anappreciation of the real exchange rate. The exchange rate apprecia-tion causes consumer price inflation to fall in the near term, althoughhigher demand pressures eventually push up domestic goods pricesby enough to cause consumer prices to rise.

These effects are qualitatively similar in the case of an au-tonomous shock to consumption demand in FRB/Global, i.e., ashock to the statistical residual in the consumption equation that isscaled to have the same effect on consumption. But from a quan-titative perspective, the output effects are considerably larger inFRB/Global, because there is less crowding out of investment and asmaller decline in real net exports in that model (not shown). Thesmaller investment decline in FRB/Global reflects that investmentis less sensitive to the long-term real interest rate (noting that long-term rates rise by a roughly commensurate amount in each model).The smaller decline in real net exports in FRB/Global reflects sev-eral factors, including less passthrough of the exchange rate to importprices, modestly lower trade price elasticities, and somewhat smallerreal appreciation.

As in the case of the government spending rise, allowing for im-perfect information about the persistence of the taste shock allowsSIGMA to imply a larger and more persistent output response thatis very similar to that in FRB/Global. The more gradual rise inlong-term interest rates reduces the magnitude of real exchange rateappreciation, leading to a smaller export decline.

3.4 Fall in Home-Currency Risk Premium

Figure 4 shows the effects on the home country of a decline inthe risk premium (νbt) on home-currency-denominated assets. As inMcCallum and Nelson (1999) and in Kollman (2001), in this sim-ulation we shock the exogenous component of the risk premium inthe uncovered interest parity condition implied by the log-linearizedmodel. The shock is scaled so that it induces an initial real appreci-ation of 10 percent, and the persistence of the shock is ρb = 0.95.

This shock reduces the required real return on all home-currency-denominated assets relative to the return on foreign assets. Thelower required real return on home-currency assets occurs through a


Figure 4. Fall in Home-Currency Risk Premium


combination of persistently lower real interest rates, and throughexpected real currency depreciation. Thus, long-term real interestrates fall (not shown), and given that the shock has no long-runeffect on the real exchange rate, the exchange rate is required toappreciate sharply in the impact period.

The appreciation of the real exchange rate depresses real exportsand raises imports—and thus exerts a contractionary effect on realGDP. However, private domestic absorption is stimulated by lowerreal interest rates and lower import prices. As a result, real GDPshows only a modest contraction in the near term, and actuallyrises above baseline after a few years as higher investment spendingleads to progressive capital deepening. The combination of strongerdomestic demand and real exchange rate appreciation induces asignificant deterioration of the nominal trade balance exceeding11⁄2 percentage points of GDP. Finally, PCE inflation shows a siz-able but transient drop due to declining import prices.

The qualitative effects in FRB/Global are very similar, with theexchange rate appreciation leading to an initial output decline, lowerreal interest rates, some decline in inflation, and a trade balance de-terioration. But while the response of GDP is fairly similar acrossmodels, there is considerable quantitative disparity in the responsesof the expenditure components: private absorption rises much morein SIGMA than in FRB/Global, and real net exports correspond-ingly exhibit a larger contraction (not shown, though suggested bythe larger trade balance response). A key factor accounting for thesedifferences is that the passthrough of exchange rate changes to im-port prices is effectively 100 percent after a couple of quarters inSIGMA, whereas it is only about 30 percent in FRB/Global. Thehigher passthrough directly accounts for the larger effects on real im-ports and exports in the former model, and contributes significantlyto driving the sharper swings in PCE price inflation and privateabsorption. As noted in the discussion of the government spendingshock, the substantial response of private absorption in SIGMA alsoreflects a highly elastic response to the persistent decline in long-termreal interest rates (not shown).

SIGMA’s implication of very high exchange rate passthroughto import prices seems at odds with empirical evidence for the


United States that estimates long-run passthrough in the rangeof 20–30 percent.23 Importantly, we emphasize that any model inwhich the desired markup is fixed (as in SIGMA) would appear un-able to match the empirical pattern of import price response in-corporated into FRB/Global, in which import prices react fairlyquickly to exchange rate changes, but then adjust little subsequently.For example, the dash-dotted lines marked “sluggish import prices”use a calibration of SIGMA in which the mean duration of exportprices is eight quarters (rather than two quarters as in the baseline).Even in this case of long-lived local currency pricing, exchange ratepassthrough is nearly complete after two years, and there is muchlarger trade adjustment than in FRB/Global.

Given the central role of exchange rate passthrough in affectingthe transmission of open economy shocks, we believe that it is ofcrucial importance to develop a theoretical framework that has theflexibility to account for the empirical features of passthrough evi-dent in U.S. data. While some microfounded models can account forlong-run passthrough that is below unity, including models with adistribution sector for retail goods as in Corsetti and Dedola (2003),and Corsetti, Dedola, and Leduc (2005), such models do not allowenough variation in desired markups to come close to matching thelow level of passthrough in U.S. data.24 Without much variation inthe desired markup, models such as SIGMA imply implausibly largeexpenditure-switching effects in response to exchange rate move-ments that may limit their usefulness in addressing some importantpolicy questions. Moreover, such restrictions may lead to large biasesin the estimates of structural parameters.

3.5 Alternative Foreign Demand Shocks

Figure 5 shows the effect on the home country of a rise in foreigninvestment demand. Specifically, investment in the foreign country

23The low long-run passthrough in FRB/Global is consistent with recent em-pirical estimates for the United States; see Marazzi, Sheets, and Vigfusson (2005).

24For example, Corsetti, Dedola, and Leduc (2005) find that their inclusion ofa distribution sector can account for a reduction in long-run passthrough fromunity to around 0.9. Other frameworks, such as Bergin and Feenstra (2001) andGust and Sheets (2006) incorporate strategic complementarities in price settingà la Kimball (1995) that allow for greater variation in desired markups. Theseappear to be promising avenues to reduce long-run passthrough.


Figure 5. Rise in Foreign Investment Demand


increases due to a highly persistent decline in its capital incometax rate; but it is useful to interpret the shock more broadly asreflecting changes in the investment climate abroad that boost theperceived return to capital. The shock is scaled so that foreign outputeventually rises by 1 percent relative to baseline.

This foreign investment demand shock operates through similarchannels both in SIGMA and in FRB/Global, and has qualitativelysimilar effects in each. Thus, the rise in foreign demand stimulateshome real net exports, both directly because of the rise in foreign ab-sorption, and indirectly through a depreciation of the home country’sreal exchange rate (not shown). This raises home output, though thestimulative effect on GDP arising from higher net exports is partlyoffset by declining private absorption (as domestic interest rates rise).The rise in real net exports generates an improvement in the nom-inal trade balance, while consumer prices rise due to higher importprices and stronger activity.

From a quantitative perspective, the “spillover effects” of the for-eign demand increase on the home country are broadly similar forthe first two years following the shock across the two models. Thissuggests that SDGE models may account for substantial spillovereffects through trade channels in response to certain types of shocks,although we caution that the fact that the shock affects foreign in-vestment spending—which is heavily import intensive—plays an im-portant role in accounting for the relatively large effects. In addition,the spillover effects would decline if SIGMA incorporated lower ex-change rate passthrough, since this would diminish the magnitudeof the home country’s export improvement. It is also evident thatSIGMA implies much greater volatility in the expenditure compo-nents than FRB/Global. The trade balance exhibits a much largerimprovement, and the components of private absorption fall muchmore sharply than in FRB/Global. As might be expected, the in-clusion of imperfect information can markedly damp the volatilityof the expenditure components in SIGMA by generating a smallerand more gradual response of long-term real interest rates, and bydamping the impact on the real exchange rate.

Figure 6 illustrates that spillover effects on the home economywould be much smaller if the foreign output expansion were insteadattributable to higher consumption spending. In particular, this al-ternative simulation assumes that foreign consumption demand rises


Figure 6. Rise in Foreign Consumption Demand


due to a taste shock, with the shock again scaled so that foreign out-put peaks at 1 percent above baseline. Given that foreign investmentis crowded out by the shock, the shock has much smaller effects onthe home country’s real exports, and induces only a small and tran-sient rise in output in our benchmark calibration of SIGMA. Whileit is interesting economically that SIGMA implies that the source ofthe foreign demand shock may be quite important in determining itseffects on the domestic economy (whereas the domestic output in-crease in FRB/Global is broadly similar across the foreign demandshocks), it is plausible that certain features of SIGMA may undulyrestrict the ability of the model to generate substantial spillover ef-fects. These include both the high interest sensitivity of private ab-sorption to the long-term real interest rate, and the omission of otherpotentially important financial channels. Recent work by Gilchrist,Hairault, and Kempf (2002) suggests that a financial accelerator mayserve to enhance the ability of open economy SDGE models to ac-count for larger spillover effects.

4. Long-Run Simulations in SIGMA

We now examine our model’s implications for several supply-sideshocks. Each shock exerts a highly persistent effect on output, inpart because capital accumulation is very gradual. In addition toillustrating the model’s long-run implications for the path of adjust-ment to each of these shocks, our analysis highlights the endogenouschannels through which this adjustment occurs.

4.1 Persistent Rise in Productivity Growth

Figure 7 shows the responses of key variables to a productivitygrowth rate shock under alternative assumptions about the informa-tion structure. The shock raises technology growth by 1 percentagepoint per year over the the first five years of the simulation horizon,then decays slowly following an AR(1) process with an autocorre-lation parameter of ρp = .975. The magnitude of the technologygrowth shock is similar to that experienced in the United Statesbetween 1996 and 2000; given the decay rate, it is consistent withagents projecting GDP growth five years ahead to rise immediatelyby about 3/4 percentage point above baseline.


Figure 7. Rise in Home Productivity Growth Rate


We begin by analyzing the effects of the shock under the as-sumption that agents have full information, and hence correctlyascertain that the shock will have highly persistent effects on thefuture growth rate of productivity (see the solid lines in the figure).Households project a much sharper rise in their future income thanin the preshock baseline. This immediately stimulates consumptiondemand and depresses the saving rate. The increase in the expectedmarginal product of capital induces investment to rise (after a shortdelay). The expansion of domestic demand leads to higher real in-terest rates, putting upward pressure on the real exchange rate (notshown) and inducing a prolonged deterioration of the trade balance.

A hallmark feature of microfounded models such as SIGMA isthat they completely articulate the longer-term economic forces thatoperate to correct any “imbalances,” including in trade, its compo-nents, and the real exchange rate. The imposition of intertemporalbudget constraints (and a debt-elastic risk premium) play a key rolein generating these endogenous adjustments. In the case of the pro-ductivity acceleration, several aspects of the adjustment process ac-count for the eventual movement of the trade balance into surplus.First, there is a long-run depreciation of the real exchange rate dueto an increase in the supply of U.S. goods, which stimulates exportsand reduces imports. Second, while the saving rate declines initially,it eventually increases as current income converges toward perma-nent income. Finally, after peaking about 10 years after the initialshock, the investment rate declines as capital approaches its newlong-run level.

While these simulation results are instructive in understandingthe forces that bring about long-run adjustment, the assumption thatagents immediately recognize a productivity acceleration as perma-nent may be somewhat implausible. For example, in the U.S. expe-rience of the late 1990s, forecasts of long-term output growth (i.e.,five to ten years ahead) did not change noticeably until several yearsafter the initial rise in productivity growth. This provides some em-pirical motivation for considering a “gradual learning” case in whichagents do not immediately recognize that the productivity growthshock is highly persistent: given our calibration of the steady-stateKalman gain parameter, they initially believe the shock is mainly


transitory.25 In this case, agents project a much less pronounced risein their income profile, so that consumption jumps much less thanin the full information case; correspondingly, the figure shows thatthe consumption share of output remains nearly flat. Given reducedaggregate demand pressure relative to the full information case, realinterest rates rise much less abruptly, which accounts for the some-what larger rise in the investment share. While the trade deficitexpands, the deterioration is also somewhat muted relative to thecase of full information. This gradual adjustment of both domesticdemand and external variables in response to the shock is more inline with the U.S. experience in the late 1990s, including with therelative constancy of the saving rate during that period.

4.2 Reduction in Labor Tax Rate

As another illustration of the forces that ensure long-run adjustmentin the SIGMA model, figure 8 shows the effects of a cut in the labortax rate. The tax cut is scaled so that government’s labor tax revenuewould fall by 1 percentage point of GDP if pretax labor income andoutput were unaffected. The shock is assumed to be highly persistent,with the autoregressive parameter ρτN set to 0.975. This labor taxcut induces the fiscal deficit to rise initially by about 1 percent ofGDP and to decay slowly thereafter.26

The cut in labor taxes induces a sharp initial rise in output.The shock exerts a strong stimulative effect on aggregate demandin the short run, as the HM households immediately expand theirconsumption in response to the increase in their after-tax income.The high level of persistence of aggregate consumption reflects thatthe consumption of the HM households remains high for an ex-tended duration (given that the cut to labor taxes is very persistentand lump-sum taxes adjust slowly). Output declines from its initialpeak as higher real interest rates crowd out investment spending andthe consumption of optimizing households. However, output remainspersistently above its preshock level even in the longer term be-cause the shock exerts substantial supply-side effects: lower tax rates

25The temporary shock in this case is i.i.d., and the Kalman gain parameteron the permanent component is 0.10.

26We provide a more detailed discussion of the effects on the trade deficit inErceg, Guerrieri, and Gust (2005).


Figure 8. Fall in Home Labor Tax Rate


induce households to work more by raising the cost of leisure, andthe rise in labor supply in turn encourages capital accumulation.

A clear advantage of our framework is that it is well suited toexplore sensitivity to various structural characteristics determiningconsumption and labor supply behavior. In the context of assessingthe effects of tax cuts, it is often of interest to policymakers to as-certain how results depend on the extent to which households are“Ricardian,” or on their labor supply elasticity. We explore each ofthese alternatives in the figures. The dashed lines show the case inwhich all households are optimizers (rather than assuming 50 per-cent are optimizers as in the baseline). Because all households inter-nalize the future tax increases necessary to satisfy the governmentintertemporal budget constraint, consumption shows a much smallerinitial increase. In the longer term, the responses are similar to thebenchmark, though there is somewhat greater capital deepening inthis case. The dash-dotted line shows the case of a much higherFrisch labor supply elasticity (equal to unity, rather than 0.2 in ourbaseline). The larger labor supply response induces a much higherlevel of capital accumulation; accordingly, the longer-run output andconsumption rise is greatly accentuated relative to the benchmarkcalibration.

The implications for the real exchange rate and trade balanceunder the alternative calibrations are shown in the lower panels. Inthe benchmark, the sharp rise in the real interest rate induces aninitial appreciation in the real exchange rate, which drives the tradebalance to deteriorate by around 0.1 percentage point of GDP. Inthe longer term, the supply-side effects dominate. Thus, the realexchange rate depreciates due to the higher supply of U.S. goods,while the trade balance shifts into surplus for the same reasons as inthe case of the productivity acceleration. The trade balance exhibitsa slightly larger deterioration in the calibration with a high laborsupply elasticity (given the large stimulative effect on investment),while real exchange rates and trade show little reaction in the casein which all agents optimize.

4.3 Reduction in the Capital Tax Rate

Figure 9 shows the effects of a cut in the capital income tax rate thatis scaled so that capital tax revenue would fall by 1 percentage point


Figure 9. Fall in Home Capital Tax Rate


of GDP if pretax labor income and output were unaffected. Theshock is assumed to be highly persistent, with the autoregressiveparameter set to 0.95. The capital tax cut induces the fiscal deficitto rise initially by about 1 percent of GDP and to decay slowlythereafter.

Given that the HM agents do not pay capital taxes, the capitaltax rate reduction does not impart the same sort of short-runaggregate demand stimulus evident in the case of the labor tax cut.Accordingly, output increases slowly in line with the gradual risein the capital stock. Higher real interest rates encourage somewhatgreater saving by optimizing agents, even though the aggregate sav-ing rate responds somewhat less due to the presence of the HMagents. The sharp rise in investment and high import content of in-vestment goods encourages a substantial rise in imports, and thispressure on the trade balance is reinforced by a short-run real ex-change rate appreciation: as a result, the trade balance experiencesa peak deterioration of roughly 1/4 percentage point of GDP. In thelong run, real exchange rate depreciation, a fall in the investmentrate, and some rise in the saving rate (in part due to higher taxeson HM agents) induce the trade balance to move into persistentsurplus.

Figure 9 also shows that output and investment would exhibitlarger responses to a capital tax cut if all agents were optimizers(dotted lines). This reflects that the larger response of domestic sav-ing in the latter case reduces pressure on real interest rates. Theoutput response would be markedly accentuated if all agents wereoptimizers and had a much higher Frisch elasticity of labor supply ofunity (five times higher than in the baseline case; see the dash-dottedlines), since the larger labor supply response would encourage highercapital accumulation.

5. Conclusion

The recent surge in interest in developing SDGE models for pol-icy analysis seems warranted. The SDGE framework possesses somekey advantages over that of existing large-scale econometric modelsby providing a clear linkage between structural features of the econ-omy and its responses to shocks. Moreover, it offers a theoretically


consistent framework for analyzing both short- and long-run re-sponses that is helpful in assessing how the economy returns to itsbalanced growth path following disturbances.

While estimation remains an important future research objective,we have argued that an essential prerequisite involves identifyingtheoretical constraints of the particular SDGE framework that maypreclude fitting the data on key dimensions. In this vein, even thoughit is encouraging that SIGMA implies responses to policy shocksthat are generally similar to FRB/Global, there are at least twosalient differences that seem attributable to restrictive aspects of ourframework: namely, SIGMA implies much larger responses of bothtrade flows and the domestic expenditure components in response toshocks, and smaller and more transient spillover effects in responseto foreign disturbances. Importantly, SIGMA’s implications alongthese dimensions are likely to characterize a broad class of currentSDGE models that imply fixed (or nearly fixed) desired markups,and that adopt a fairly standard framework for modeling investmentand consumption behavior.

Given the importance of the magnitude of expenditure-switchingeffects for a wide range of open economy questions, we believe thatit will be important in future work to develop mechanisms that canaccount for much lower long-term passthrough than implied by ourmodel. Moreover, it will also be desirable to incorporate features thatcan potentially account for larger responses to foreign disturbances,at least under some conditions. It seems plausible that allowing forsectoral attachments of factors of production or the inclusion of afinancial accelerator may allow SDGE models greater flexibility onthis dimension.

Finally, while comparisons with FRB/Global are useful in eval-uating the flexibility of SIGMA to fit responses similar to that of adata-oriented econometric model, we intend to adopt an estimationstrategy that would allow significant departures from the responsesof FRB/Global if the data provide a strong enough rationale. Inparticular, FRB/Global responses could be helpful in setting priorsover certain estimated parameters in SIGMA in the context of aBayesian approach, though sizable differences could emerge betweenthe responses of the two models if the posterior distribution over theparameters diverged significantly from the prior.


References

Amato, Jeffery D., and Thomas Laubach. 2003. “Estimation andControl of an Optimization-Based Model with Sticky Prices andWages.” Journal of Economic Dynamics and Control 27 (7):1181–1215.

Anderson, Gary. 1997. “A Reliable and Computationally EfficientAlgorithm for Imposing the Saddle Point Property in DynamicModels.” Occasional Staff’s Studies 4, Board of Governors of theFederal Reserve System.

Anderson, Gary, and George Moore. 1985. “A Linear Algebraic Pro-cedure for Solving Linear Perfect Foresight Models.” EconomicLetters 17 (3): 247–52.

Bergin, Paul R., and Robert C. Feenstra. 2001. “Pricing-to-Market,Staggered Contracts, and Real Exchange Rate Persistence.”Journal of International Economics 54 (2): 333–59.

Betts, Caroline, and Michael B. Devereux. 1996. “The ExchangeRate in a Model of Pricing-to-Market.” European EconomicReview 40 (3–5): 1007–21.

Blanchard, Olivier J., and C. M. Kahn. 1980. “The Solution of LinearDifference Models under Rational Expectations.” Econometrica48 (5): 1305–12.

Blanchard, Olivier J., and Roberto Perotti. 2002. “An EmpiricalCharacterization of the Dynamic Effects of Changes in Gov-ernment Spending and Taxes on Output.” Quarterly Journal ofEconomics 117 (4): 1329–68.

Brayton, Flint, Andrew Levin, Ralph Tryon, and John C. Williams.1997. “The Evolution of Macro Models at the Federal ReserveBoard.” Finance and Economics Discussion Series, No. 1997-29,Board of Governors of the Federal Reserve System.

Brayton, Flint, and Peter Tinsley. 1996. “A Guide to FRB/US.”Finance and Economics Discussion Series, No. 1996-42, Board ofGovernors of the Federal Reserve System.

Calvo, Guillermo A. 1983. “Staggered Prices in a Utility-MaximizingFramework.” Journal of Monetary Economics 12 (3): 383–98.

Campa, José, and Linda Goldberg. 2004. “Exchange Rate Pass-Through into Import Prices.” CEPR Discussion Paper No. 4391.

Christiano, Lawrence J., Martin Eichenbaum, and Charles L. Evans.2005. “Nominal Rigidities and the Dynamic Effects of a Shock


to Monetary Policy.” Journal of Political Economy 113 (1):1–45.

Corsetti, Giancarlo, and Luca Dedola. 2003. “Macroeconomics ofInternational Price Discrimination.” CEPR Discussion PaperNo. 3710.

Corsetti, Giancarlo, Luca Dedola, and Sylvain Leduc. 2005. “DSGEModels of High Exchange-Rate Volatility and Low Pass-Through.�

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